Question 175
Find the value of
$$
\sum_{1 \leq a \lt b \lt c} \frac{1}{2^{a} 3^{b} 5^{c}}
$$
(i.e. the sum of $\frac{1}{2^{a} 3^{b} 5^{c}}$ over all triples of positive integers $(a, b, c)$ satisfying $a \lt b \lt c$).
$$
\sum_{1 \leq a \lt b \lt c} \frac{1}{2^{a} 3^{b} 5^{c}}
$$
(i.e. the sum of $\frac{1}{2^{a} 3^{b} 5^{c}}$ over all triples of positive integers $(a, b, c)$ satisfying $a \lt b \lt c$).